Washington (CNN) -- The problematic intersection of health care and abortion politics will be highlighted again Monday as religious abortion rights supporters demand changes to reform legislation recently passed by the House of Representatives.

Members of the Religious Coalition for Reproductive Choice will hold a news conference calling on the Senate to alter language in the House bill that places explicit restrictions on federal funding for abortion.

"Our health care system should be inclusive and respectful of diverse religious beliefs and decisions regarding childbearing," the group said in a statement.

"A health care system that serves all persons with dignity and equality will include comprehensive reproductive health services."

The coalition is an umbrella organization comprised of representatives of a variety of progressive Catholic, Protestant, Jewish and other groups.

Religious leaders have been split over the issue of abortion coverage. The U.S. Conference of Catholic Bishops pushed successfully to amend the House bill to prohibit abortion coverage in a government-run health insurance plan, as well as in private plans that accept anyone using government subsidies to buy insurance coverage.

Current law restricts federal funding for abortions to cases of rape, incest or to save the life of the mother.

Under the House bill, people would be permitted to buy supplemental coverage for abortions with their own money. Abortion rights groups have called the idea discriminatory and preposterous, arguing that women rarely plan ahead for an abortion.

"Health care reform must not be misused as an opportunity to restrict women's access to reproductive health services," 90 House Democrats opposed to the amended abortion language wrote in a letter to President Obama last week.

In an example of the complicated politics of the issue, all but one of the House Democrats who signed the letter had voted to pass the overall health care bill even though it contained the amendment they opposed. Their strategy now is to work with Obama and Senate Democrats to prevent inclusion of the language of the amendment from a final health care bill.

In an interview broadcast Sunday on CNN's "State of the Union," presidential adviser David Axelrod said the House bill as currently worded goes further than Obama's stated wishes.

"The president has said repeatedly, and he said in his speech to Congress, that he doesn't believe that this bill should change the status quo as it relates to the issue of abortion," Axelrod said.

However, Axelrod dodged the question of whether Obama would veto a final bill that included the same language as the House bill.

Anti-abortion Senate Democrats say they will seek strict funding restrictions for abortion when the Senate debates its own version of the bill in coming weeks.

"Now, whether the House formula has got it quite right or not, that's open to debate," Sen. Kent Conrad of North Dakota said Sunday on CNN. "But what is clear is, to have legislation passed -- it was clear in the House; it will be clear in the Senate -- there cannot be taxpayer funding of abortion."